For a connected graph G = (V,E) of order at least two, a total detour monophonic set of a graph G is a detour monophonic set S such that the subgraph induced by S has no isolated vertices. The minimum cardinality of a total detour monophonic set of G is the total detour monophonic number of G and is denoted by dmt(G). A subset T of a minimum total detour monophonic set S of G is a forcing total...

For vertices x and y in a connected graph G, the detour distance D(x, y) is the length of a longest x− y path in G. An x− y path of length D(x, y) is an x − y detour. The closed detour interval ID[x, y] consists of x, y, and all vertices lying on some x − y detour of G; while for S ⊆ V (G), ID[S] = ⋃ x,y∈S ID[x, y]. A set S of vertices is a detour convex set if ID[S] = S. The detour convex hull...

For any vertex x in a connected graph G of order p ≥ 2, a set S of vertices of V is an x-detour set of G if each vertex v in G lies on an x-y detour for some element y in S. A connected x-detour set of G is an x-detour set S such that the subgraph G[S] induced by S is connected. The minimum cardinality of a connected x-detour set of G is the connected x-detour number of G and is denoted by cdx(...

In the present study, four detour matrix-based Topological Indices (TIs) termed as augmented path eccentric connectivity indices 1-4 (denoted by (AP)ξ(1)(C), (AP)ξ(2)(C), (AP)ξ(3)(C) and (AP)ξ(4)(C)) as well as their topochemical versions (denoted by (AP)ξ(1c)(C), (AP)ξ(2c)(C), (AP)ξ(3c)(C) and (AP)ξ(4c)(C)) have been conceptualised. A modified detour matrix termed as chemical detour matrix (Δ(...

For a connected graph G of order at least two, a path P is called a monophonic path if it is a chordless path. A longest x−y monophonic path is called an x − y detour monophonic path. A set S of vertices of G is an edge detour monophonic set of G if every edge of G lies on a detour monophonic path joining some pair of vertices in S. The edge detour monophonic number of G is the minimum cardinal...

A vertex v in a connected fuzzy graph G is a fuzzy detour boundary vertex of a vertex u, if ) , ( ) , ( v u w u Δ ≤ Δ for every fuzzy detour neighbour w of v. This paper is devoted to see how the fuzzy detour boundary vertices are related to fuzzy detour interior vertices. Further, the relation between the fuzzy detour boundary vertices and fuzzy detour eccentric vertices are also studied. Some...

Relative centricity RC values of vertices/atoms are calculated within the Distance Detour and Cluj-Distance criteria on their corresponding Shell transforms. The vertex RC distribution in a molecular graph gives atom equivalence classes, useful in interpretation of NMR spectra. Timed by vertex valences, RC provides a new index, called Centric Connectivity CC, which can be useful in the topologi...

For two vertices u and v in a graph G = (V, E), the detour distance D (u, v) is the length of a longest u – v path in G. A u – v path of length D (u, v) is called a u – v detour. For subsets A and B of V, the detour distance D (A, B) is defined as D (A, B) = min {D (x, y) : x ∈ A, y ∈ B}. A u – v path of length D (A, B) is called an A – B detour joining the sets A, B V where u ∈ A and v ∈ B. A...

For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ V(G) is an x-detour set of G if each vertex v ∈ V(G) lies on an x − y detour for some element y in S. The minimum cardinality of an xdetour set of G is defined as the x-detour number of G, denoted by dx(G). An x-detour set of cardinality dx(G) is called a dx-set of G. A connected x-detour set of G is an x-detour set S such th...